-
Previous Article
Sharp interface limit of the Fisher-KPP equation when initial data have slow exponential decay
- DCDS-B Home
- This Issue
- Next Article
The Euler-Maruyama approximations for the CEV model
1. | School of Mathematical Sciences, Monash University, Clayton Campus, Building 28, Wellington road, Victoria, 3800, Australia |
2. | School of Mathematical Sciences, Monash University, Clayton Campus, Building 28,, Wellington road, Victoria, 3800, Australia |
3. | Department of Engineering Systems, Tel Aviv University, Tel Aviv, Ramat Aviv, 69978, Israel |
References:
[1] |
A. Alfonsi, Higher order discretization schemes for the CIR process: application to affine term structure and Heston models,, Mathematics of Computation, 79 (2010), 209.
doi: 10.1090/S0025-5718-09-02252-2. |
[2] |
V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations I. Convergence rate of the distribution function,, Probability Theory and Related Fields, 104 (1996), 43.
doi: 10.1007/BF01303802. |
[3] |
P. Billingsley, "Converges of Probability Measures,", John Wiley & Sons, (1968).
|
[4] |
M. Bossy and A. Diop, An efficient discretisation scheme for one dimensional SDEs with a diffusion coefficient function of the form $|x|^\alpha, \alpha\in [1/2, 1)$,, preprint, (2007). Google Scholar |
[5] |
J. C. Cox, The constant elasticity of variance option. Pricing model,, The Journal of Portfolio Management, 23 (1997), 15. Google Scholar |
[6] |
D. Dawson, http://www.math.ubc.ca/ db5d/SummerSchool09/lectures-dd/lecture4.pdf,, (Accessed on August 3, (2010). Google Scholar |
[7] |
G. Deelstra and F. Delbaen, Convergence of discretized stochastic (interest rate) processes with stochastic drift term,, Applied Stochastic Models and Data Analysis, 14 (1998), 77.
doi: 10.1002/(SICI)1099-0747(199803)14:1<77::AID-ASM338>3.0.CO;2-2. |
[8] |
F. Delbaen and H. A. Shirakawa, Note of option pricing for constant elasticity of variance model,, Asia-Pacific Financial Markets, 9 (2002), 159.
doi: 10.1023/A:1024173029378. |
[9] |
W. Feller, Two singular diffusion problems,, Annals of Mathematics, 54 (1951), 173.
doi: 10.2307/1969318. |
[10] |
I. Gy, A note on Eulers approximations,, Potential Analysis, 8 (1998), 205.
doi: 10.1023/A:1008605221617. |
[11] |
I. Gy,I. ongy and N. Krylov, Existence of strong solutions for Itó's stochastic equations via approximations,, Probability Theory and Related Fields, 105 (1996), 143.
doi: 10.1007/BF01203833. |
[12] |
N. Halidias and P. Kloeden, A note on the Euler-Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient,, BIT Numerical Mathematics, 48 (2008), 51.
doi: 10.1007/s10543-008-0164-1. |
[13] |
P. L. Hennequin and A. Tortrat, "Théorie des Probabilités et Quelques Applications,", Masson et Cie, (1965).
|
[14] |
D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the mean-reverting square root process,, Computational Finance, 8 (2005), 35. Google Scholar |
[15] |
D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations,, SIAM Journal of Numerical Analysis, 40 (2002), 1041.
doi: 10.1137/S0036142901389530. |
[16] |
M. Hutzenthaler and A. Jentzen, Non-globally Lipschitz counterexamples for the stochastic Euler scheme,, preprint, (). Google Scholar |
[17] |
A. Jentzen, P. E. Kloeden and A. Neuenkirch, Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coeffcients,, Numerische Mathematik, 112 (2009), 41.
doi: 10.1007/s00211-008-0200-8. |
[18] |
Yu. Kabanov, R. Liptser and A. N. Shiryaev, Estimates of closeness in variation of probability measures (Russian),, Dokl. Akad. Nauk SSSR, 278 (1984), 265.
|
[19] |
F. C. Klebaner, "Introduction to Stochastic Calculus with Applications,", 2$^{nd}$ edition, (2005).
|
[20] |
P. E. Kloeden and K. Platten, "Numerical Solution of Stochastic Differential Equations,", Springer-Verlag, (1992).
|
[21] |
R. Sh. Liptser and A. N. Shiryayev, "Theory of Martingales" [Translated from the Russian by K. Dzjaparidze], Mathematics and its Applications (Soviet Series), 49 (1989).
|
[22] |
G. N. Milstein and M. V. Tretyakov, "Stochastic Numerics for Mathematical Physics,", Springer-Verlag, (2004).
|
[23] |
L. C. G. Rogers and D. Williams, "Diffusions, Markov Processes, and Martingales. Vol. 2. Itô Calculus,", Reprint of the 2$^{nd}$ edition, (2000).
|
[24] |
T. Shiga and S. Watanabe, Bessel diffusions as a one parameter family of diffusion processes,, Zeitschrift f\, 27 (1973), 37.
|
[25] |
D. Talay, Simulation and numerical analysis of stochastic differential systems: A review,, in, 451 (1995), 63. Google Scholar |
[26] |
L. Yan, The Euler scheme with irregular coefficients,, Annals of Probability, 30 (2002), 1172.
doi: 10.1214/aop/1029867124. |
[27] |
H. Zähle, Weak approximation of SDEs by discrete time processes,, Journal of Applied Mathematics and Stochastic Analysis, 2008 (2008).
|
show all references
References:
[1] |
A. Alfonsi, Higher order discretization schemes for the CIR process: application to affine term structure and Heston models,, Mathematics of Computation, 79 (2010), 209.
doi: 10.1090/S0025-5718-09-02252-2. |
[2] |
V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations I. Convergence rate of the distribution function,, Probability Theory and Related Fields, 104 (1996), 43.
doi: 10.1007/BF01303802. |
[3] |
P. Billingsley, "Converges of Probability Measures,", John Wiley & Sons, (1968).
|
[4] |
M. Bossy and A. Diop, An efficient discretisation scheme for one dimensional SDEs with a diffusion coefficient function of the form $|x|^\alpha, \alpha\in [1/2, 1)$,, preprint, (2007). Google Scholar |
[5] |
J. C. Cox, The constant elasticity of variance option. Pricing model,, The Journal of Portfolio Management, 23 (1997), 15. Google Scholar |
[6] |
D. Dawson, http://www.math.ubc.ca/ db5d/SummerSchool09/lectures-dd/lecture4.pdf,, (Accessed on August 3, (2010). Google Scholar |
[7] |
G. Deelstra and F. Delbaen, Convergence of discretized stochastic (interest rate) processes with stochastic drift term,, Applied Stochastic Models and Data Analysis, 14 (1998), 77.
doi: 10.1002/(SICI)1099-0747(199803)14:1<77::AID-ASM338>3.0.CO;2-2. |
[8] |
F. Delbaen and H. A. Shirakawa, Note of option pricing for constant elasticity of variance model,, Asia-Pacific Financial Markets, 9 (2002), 159.
doi: 10.1023/A:1024173029378. |
[9] |
W. Feller, Two singular diffusion problems,, Annals of Mathematics, 54 (1951), 173.
doi: 10.2307/1969318. |
[10] |
I. Gy, A note on Eulers approximations,, Potential Analysis, 8 (1998), 205.
doi: 10.1023/A:1008605221617. |
[11] |
I. Gy,I. ongy and N. Krylov, Existence of strong solutions for Itó's stochastic equations via approximations,, Probability Theory and Related Fields, 105 (1996), 143.
doi: 10.1007/BF01203833. |
[12] |
N. Halidias and P. Kloeden, A note on the Euler-Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient,, BIT Numerical Mathematics, 48 (2008), 51.
doi: 10.1007/s10543-008-0164-1. |
[13] |
P. L. Hennequin and A. Tortrat, "Théorie des Probabilités et Quelques Applications,", Masson et Cie, (1965).
|
[14] |
D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the mean-reverting square root process,, Computational Finance, 8 (2005), 35. Google Scholar |
[15] |
D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations,, SIAM Journal of Numerical Analysis, 40 (2002), 1041.
doi: 10.1137/S0036142901389530. |
[16] |
M. Hutzenthaler and A. Jentzen, Non-globally Lipschitz counterexamples for the stochastic Euler scheme,, preprint, (). Google Scholar |
[17] |
A. Jentzen, P. E. Kloeden and A. Neuenkirch, Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coeffcients,, Numerische Mathematik, 112 (2009), 41.
doi: 10.1007/s00211-008-0200-8. |
[18] |
Yu. Kabanov, R. Liptser and A. N. Shiryaev, Estimates of closeness in variation of probability measures (Russian),, Dokl. Akad. Nauk SSSR, 278 (1984), 265.
|
[19] |
F. C. Klebaner, "Introduction to Stochastic Calculus with Applications,", 2$^{nd}$ edition, (2005).
|
[20] |
P. E. Kloeden and K. Platten, "Numerical Solution of Stochastic Differential Equations,", Springer-Verlag, (1992).
|
[21] |
R. Sh. Liptser and A. N. Shiryayev, "Theory of Martingales" [Translated from the Russian by K. Dzjaparidze], Mathematics and its Applications (Soviet Series), 49 (1989).
|
[22] |
G. N. Milstein and M. V. Tretyakov, "Stochastic Numerics for Mathematical Physics,", Springer-Verlag, (2004).
|
[23] |
L. C. G. Rogers and D. Williams, "Diffusions, Markov Processes, and Martingales. Vol. 2. Itô Calculus,", Reprint of the 2$^{nd}$ edition, (2000).
|
[24] |
T. Shiga and S. Watanabe, Bessel diffusions as a one parameter family of diffusion processes,, Zeitschrift f\, 27 (1973), 37.
|
[25] |
D. Talay, Simulation and numerical analysis of stochastic differential systems: A review,, in, 451 (1995), 63. Google Scholar |
[26] |
L. Yan, The Euler scheme with irregular coefficients,, Annals of Probability, 30 (2002), 1172.
doi: 10.1214/aop/1029867124. |
[27] |
H. Zähle, Weak approximation of SDEs by discrete time processes,, Journal of Applied Mathematics and Stochastic Analysis, 2008 (2008).
|
[1] |
Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366 |
[2] |
Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331 |
[3] |
Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325 |
[4] |
Xiuli Xu, Xueke Pu. Optimal convergence rates of the magnetohydrodynamic model for quantum plasmas with potential force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 987-1010. doi: 10.3934/dcdsb.2020150 |
[5] |
Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226 |
[6] |
Chungang Shi, Wei Wang, Dafeng Chen. Weak time discretization for slow-fast stochastic reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021019 |
[7] |
Thomas Frenzel, Matthias Liero. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345 |
[8] |
Gheorghe Craciun, Jiaxin Jin, Casian Pantea, Adrian Tudorascu. Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1305-1335. doi: 10.3934/dcdsb.2020164 |
[9] |
Ziang Long, Penghang Yin, Jack Xin. Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data. Inverse Problems & Imaging, 2021, 15 (1) : 41-62. doi: 10.3934/ipi.2020077 |
[10] |
Nalin Fonseka, Jerome Goddard II, Ratnasingham Shivaji, Byungjae Son. A diffusive weak Allee effect model with U-shaped emigration and matrix hostility. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020356 |
[11] |
Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304 |
[12] |
Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242 |
[13] |
Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020400 |
[14] |
Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021004 |
[15] |
Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020316 |
[16] |
Yoshihisa Morita, Kunimochi Sakamoto. Turing type instability in a diffusion model with mass transport on the boundary. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3813-3836. doi: 10.3934/dcds.2020160 |
[17] |
Zhihua Liu, Yayun Wu, Xiangming Zhang. Existence of periodic wave trains for an age-structured model with diffusion. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021009 |
[18] |
Hui Zhao, Zhengrong Liu, Yiren Chen. Global dynamics of a chemotaxis model with signal-dependent diffusion and sensitivity. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021011 |
[19] |
Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021017 |
[20] |
Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]